# Tutorial Track

## J. Cummings: Singular cardinal combinatorics

The tutorial will focus on the combinatorics of singular cardinals and their successors, with a special emphasis on compactness phenomena.

## M. Hušek: Large cardinals in general topology

Influence of some large cardinals on results in topological structures will be discussed. Mainly Ulam measurable cardinal, sequential cardinal and their "higher" concepts (measurable and submeasurable cardinals) will be considered. For topological structures we take topological spaces, uniform spaces, topological groups, topological linear spaces and their nice subcategories. Among the properties discussed are commutations of extensions with products, simplicity of some classes of structures, sequential continuity versus continuity and productivity of special classes of structures. One of the main tools is factorization of mappings defined in products. In some factorizations $\lambda$-strongly compact cardinals will be used.

## W. Kubiś: Generic structures

Generic structures are mathematical objects with rich automorphism groups, arising as limits of special sequences. These objects can also be described by the existence of a winning strategy in a natural infinite game where two players alternately choose bigger and bigger structures of the same type. The theory of generic structures originates from the work of Roland Fraı̈ssé in 1954 and for a long time it was considered to be part of model theory. On the other hand, one of the first generic structures was discovered by Urysohn: A universal ultrahomogeneous Polish metric space. As it happens, generic objects give rise to examples of Polish groups with very interesting dynamics.

The purpose of this tutorial is to present the theory of generic structures, with selected applications and examples from several areas of mathematics.

## J. Lopez-Abad: Extremely amenable automorphism groups

The goal of this mini course is to present the main points in the study of the topological dynamics of topological groups when represented as automorphism groups of approximate ultrahomogeneous metric structures. More particularly, we will discuss:

• Fraïssé and metric Fraïssé correspondence for (approximate) ultrahomogeneous structures (AuH).
• Representation of polish groups as automorphism groups $\mathrm{Aut}(\mathcal M)$ of (AuH) structures $\mathcal M$.
• The Kechris–Pestov–Todorcevic (KPT) correspondence between extremely amenable groups of automorphisms of (AuH) structures and the approximate Ramsey property.
• The computation of universal minimal flows.
• Examples, including
1. $\mathrm{Aut}(\mathbb Q,<)$,
2. $\mathrm{Aut}(\mathbb B)$, where $\mathbb B$ is the countable atomless boolean algebra,
3. $\mathrm{Aut}(\mathbb G)$, where $\mathbb G$ is the Gurarij space,
4. $\mathrm{Aut}(L_p[0,1])$.

# Research Track

SpeakerTitleAbstract/Slides
Viera ŠottováThe first sketch: $\mathcal{N}$ with respect to ideals on $\omega$ slides
Szymon ŻeberskiSuperperfect Mycielski slides
Serhii BardylaDense and closed subsets of compact-like topological spacesabstract slides
Adam BartošCrookedness and almost homogeneity in categories of compactaabstract slides
Judyta BąkOn the productivity of the countable chain condition slides
Mariam BeriashviliApplications of Mazurkiewicz type sets in the study of measurability properties of sets and functionsabstract slides
José Antonio CoronaOpen coloringsabstract slides
Cesar CorralUniformization properties of ladder systems under MA(S)abstract slides
Matheus Duzi Ferreira CostaTopological Games of Bounded Selectionsabstract slides
Mirna DzamonjaLogics compact at singular cardinals slides
Monroe EskewElementary embeddings and correctnessabstract slides
Miha HabičSome results on ultrapower capturingabstract slides
K P HartYou Chooseabstract slides
Radek HonzikProjective wellorder and the tree propertyabstract slides
Eliza JabłońskaOn $n$-fold sum of a non-flat continuumabstract slides
Michał KorchGeneralized Egorov's statement for idealsabstract slides
Ziemowit KostanaOn countably saturated linear ordersabstract slides
Andrzej KucharskiOn some properties of a base.abstract slides
Ashutosh KumarOrder dimension of Turing degrees slides
Borisa KuzeljevicA long chain of P-pointsabstract slides
Arkady LeidermanQuotients of free topological groupsabstract slides
Marcin MichalskiSmital and Fubini among treesabstract slides
Kaethe MindenMaximality and Resurrectionabstract slides
Sonia Navarro FloresBorel ideals and Ramsey spacesabstract slides
José de Jesús Pelayo GómezBetween Hausdorff and Selective slides
Veronika PitrováHereditary (bi)coreflective subcategories in certain categories of semitopological groupsabstract slides
Grzegorz PlebanekSmall almost disjoint families and their applicationsabstract slides
Márk PoórSets on the plane meeting every line in a set of measure 1abstract slides
Robert RalowskiContinunous images of Bernstein sets slides
Jonathan SchilhanDefinability aspects of ultrafilters slides
Damian SobotaOn the Josefson-Nissenzweig theorem for C(K)-spacesabstract slides
Šárka StejskalováIndestructibility of the tree propertyabstract slides
Jaroslav SupinaOn a level measure slides
Jarosław SwaczynaHaar-small sets in Polish groups slides
Dorottya SzirákiThe Open Dihypergraph Dichotomy for Definable Subsets of Generalized Baire Spacesabstract slides
Alessandro VignatiHeavy liftingabstract slides
Magdalena WłudeckaProducts of gamma setsabstract slides
Thilo WeinertPolarised Partition Relations for Order Typesabstract slides
Wolfgang WohofskyA Sacks amoeba preserving distributivity of $P(\omega)/\mathrm{fin}$ slides
Rafael ZamoraGenerics in a measurable wreath productabstract slides